Bevel mess over UV Sphere
A subdivision approach would be to let Catmull-Clark do your bevelling

Start with a level of subdivision lower than the one you want to wind up with, and cheat if you can: choose the number of segments in your sphere to fit your chimney profile nicely. Retain a highly-subdivided copy of the sphere in the same place, as a future Shrinkwrap target. Hide it if you like.

Create a 6 or 8-sided circle to be the profile of the chimney, raise it above the sphere, position and orient the profile over the sphere



Knife-Project the circle down on to the sphere



X Delete the faces created by the projection, and ShiftD copy the edges on the perimeter of the hole: raise them in Z and scale to 0 in Z to flatten. Select all the vertices on the surface of the sphere, and create a vertex group from them. Assign a Shrinkwrap modifier to that group, with the intact subdivided copy of the sphere as target.

J or K Cut some edges in the sphere joining the 'loose' vertices of the hole to existing sphere vertices, creating convex polygons, not too skinny or irregular, but don't worry about making all quads.. we're going to let the first level of subdivision do that for us.



CtrlE > Bridge Edge Loops between the hole and the profile, and Catmull-Clark subdivide the whole model once with a Subdivision modifier under the shrinkwrap. We're almost done.  (You can use Loop Tools > Circle to make the profile regular.) If you apply this first level of subdivision, you're all quads. Selecting loops to slide them around becomes easier.  



The rest is tweaking, smoothing, Because of the Shrinkwrap, you can move vertices around on the surface of the sphere quite a bit without disturbing its curvature. If you were really fussy, you could create a shrinkwrap target for the chimney, too. The sharpness of the bevel can be altered by sliding / adding / removing edge loops in its vicinity.